Proceedings of the 5rd International Conference on Civil Structural and Transportation Engineering (ICCSTE'20)

Niagara Falls, Canada Virtual Conference – November, 2020

Paper No. 326

DOI: 10.11159/iccste20.326

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Simulation of Spatially Varying Ground Motions at Site with Stochastic Soil Layers: A Case Study in Nw Algeria

Karim Afif Chaouch Ecole Polytechnique d'Architecture et d'Urbanisme

Route de beaulieu, B.P N°177, 16200, El Harrach, Algiers, Algeria

karim.afifchaouch@gmail.com

National Polytechnic School, Seismic Engineering and Structural Dynamics Laboratory

10 Rue des Frères OUDEK, 16200, El Harrach

Algiers, Algeria

karim.afifchaouch@gmail.com

Abstract - Seismic design of common engineering structures is based on the assumption that excitations at all support points are uniform

or fully coherent. However, in lifeline structures with dimensions of the order of wavelengths of incident seismic waves, spatial variation

of seismic ground motions caused by incoherence effects and stochasticity in the characteristics of the surface layers may introduce

significant additional forces in their structural elements. The physical characterization of the seismic spatial variation and the realistic

simulation of the spatially variable ground motion random field become then vital to the success of performance-based design of these

extended structures. This paper presents simulation of spatially varying earthquake motions at the free surface of a homogeneous layered

stochastic soil site in the epicentral region of the El-Asnam Earthquake in Algeria, considering incoherence, wave passage and site effects.

Lagged coherency functions at bedrock have been estimated by simulating ground motion field (around Sogedia Factory) corresponding

to the 1980 El-Asnam Earthquake. Considering randomness in the thickness of soil layers overlying the bedrock in the study area, an

analytical approach has been used to evaluate lagged coherency functions at the surface. Soil proprieties are assumed to vary laterally

with a Gaussian distribution. The critical parameter controlling wave passage effect is the apparent propagation velocity of shear waves

across the array and is estimated by using frequency–wave number spectrum analysis. It is observed that the simulated spatially variable

surface ground-motion time histories at different locations are compatible with the properties of the target (predictive) random field, i.e.

assumption of site homogeneity by conserving the same energy as the reference surface motion and its model of spatial coherency

function.

Keywords: Ground motion simulation; ground motion spatial variation; random soil

1. Introduction Spatially varying ground motions (SVGMs), exhibiting spatial differences in their amplitude and phases, have a

significant effect on the responses of spatially extended structures (e.g., Walling and Abrahamson [1]), such as Nuclear

Power Plant (NPP) structures, where the nonlinear structural response history analysis necessitates the synthesis of SVGMs,

as well as other lifeline structures. In order to consider such effects, time domain dynamic analysis is increasingly performed

for the seismic design of extended structures. As a result, spatially variable ground motion time histories are required as an

input at different locations along the structure. Efficiently and rationally generating asynchronous motions is an important

issue in modern earthquake engineering practice. Several stochastic methods of simulation, unconditional or conditional,

have been developed to generate spatially correlated ground motions given an empirical coherency function (see, for

example, [2] and [3]).

Hao’s method [2], one of the pioneering methods in the field and which represents a significant contribution to modern

earthquake engineering, has certain limitations that were identified, including: (1) the lack of energy conservation of

generated motions across stations, (2) distortion of generated motion due to wave passage effect only, and (3) the inability

to use general forms of incoherence functions. The method proposed by Abrahamson [3] overcomes these limitations and is

thus more suitable for use in generation of incoherent ground motion for the nonlinear dynamic analysis of extended

structures especially where general forms of incoherence functions are employed.

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The spatial variation of seismic motions generated at multiple stations is controlled by two major functions, coherency

and apparent velocity. Several generic models of lagged coherency (random phase variability), which can be classified as

empirical, semi-empirical, analytical, and theoretical models, have been proposed in the literature. Most of the coherency

models used in these methods of simulations are derived or regressed from strong motion data recorded by dense arrays such

as the El-Centro array in California, the SMART-1 strong motion array in Taiwan, and the Parkway Valley array in New

Zealand (see, for example, [4]). It is well known that coherency models calibrated from data collected in one region may not

be suitable for use in other areas (see, for example, [5]; [6]; [7]; [8]). Despite this, due to lack of local data, which is the case

in the current study area (epicentral region of the 1980 El-Asnam Earthquake in northwest Algeria), coherency models

calibrated for one region are often used to simulate ground motion in other regions, sometimes with different tectonic and

geological settings. Also, cconsequently of these difficulties and in the absence of the records, the apparent propagation

velocity of the motions is given some approximate values (ranging from low values of a few hundred to infinity), and the

excitations are assumed, as for uniform sites, to propagate from one end of the bridge structure to the other. Clearly, this may

lead to a significant variability in the response of structures, particularly the ones located at sites with soft soil conditions,

due to the significant uncertainty in the evaluation of this quantity.

In this context, AfifChaouch et al. [9] presented an extension of the finite-source Empirical Green’s Function (EGF)

method of Irikura et al. [10] to synthesize SVGMs and thereby estimate lagged coherency functions at the bedrock of the

study area. Based on simulated motion, a parametric model of lagged coherency at bedrock was also presented by

AfifChaouch et al. [9]. Then, AfifChaouch et al. [11] extended the work presented in [9] using the method of Zerva and

Harada [12] to investigate the effects of lateral soil hererogeneity in lagged coherency of strong ground motion for the case

of the Sogedia site in El-Asnam city located in the epicentral region of the 1980 El-Asnam Earthquake. Due to the lack of

sufficient information to describe lateral variability soil mechanical properties, only randomness in the thickness of the

different layers were modelled. They thus obtained the total lagged coherency functions at the ground surface, which can

now be used to simulate spatially variable surface motion for engineering applications.

In this paper, we continue their study presented in [11] to simulate spatially variable ground motions at the surface of

the same random site of Sogedia using the conditional simulation method of Abrahamson [3]. For this, the total lagged

coherency function obtained in [11] is considered as modelling the incoherence effect and, the wave passage effect is

modelled here by an apparent wave velocity estimated by using frequency–wave number spectrum analysis.

2. Spatial Coherency Function At Homogeneous Stochastic Horizontal Ground Spatial variability of ground motion is caused by a number of factors (Der Kiureghian [13]). Assuming homogenous site

conditions, SVGM are caused by the wave passage effect and incoherence effect. The wave passage is time delays in wave

arrivals due to inclined vertically propagating plane waves or horizontally propagating surface waves. The incoherence effect

is caused by complex waveform scattering occurs as the seismically generated body waves encounter heterogeneities along

their source-to-site travel path.

Consider stochastic soil layers resting on rigid bedrock and subjected to earthquake ground motion, the complex

coherency function of surface motion is composed of terms corresponding to wave passage effects, bedrock motion loss of

coherency effects, and site response contribution [12], expressed as (see, for example, [12]; [13]):

𝛾(𝜉, 𝜔) = 𝛾𝑏,𝑐𝑜ℎ(𝜉, 𝜔) ∙ 𝛾𝑏,𝑝𝑟𝑜𝑝(𝜉, 𝜔) ∙ 𝛾𝑙,𝑐𝑜ℎ(𝜉, 𝜔) (1)

where 𝛾𝑏,𝑝𝑟𝑜𝑝(𝜉, 𝜔) is lagged coherency function of bedrock motion. In this work, this function is parametrized by the

following Hindy and Novak model [14] for which the parameters were calibrated from lagged coherency functions estimated

from ground motion simulated at the bedrock of the epicentral region of the 1980 El-Asnam Earthquake by the widely used

finite-source ground motion simulation approach, the so-called Empirical Green’s Function (EGF) method.

𝛾𝑏,𝑐𝑜ℎ(𝜉, 𝜔) = 𝑒𝑥𝑝{−(𝛼𝜔𝜉)𝜆} (2)

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The 𝛾𝑙,𝑐𝑜ℎ(𝜉, 𝜔) represents the contribution of the soil stochasticity to the lagged coherency of surface motions, and is

given by [11]:

𝛾𝑙,𝑐𝑜ℎ =𝐻1(𝛽, 𝜔0, 𝜁0, 𝜔) + 𝑅𝜔𝜔(𝜉) ∙ 𝐻2(𝛽, 𝜔0, 𝜁0, 𝜔)

𝐻1(𝛽, 𝜔0, 𝜁0, 𝜔) + 𝜎𝜔𝜔2 (𝜉) ∙ 𝐻2(𝛽, 𝜔0, 𝜁0, 𝜔)

(3)

where the participation factor 𝛽, the mean value 𝜔0 and standard deviations 𝜎𝜔𝜔 and autocorrelation 𝑅𝜔𝜔 of the

predominant natural frequency of the ground, and its equivalent damping ratio 𝜁0 are all the parameters of the proposed

model of Zerva and Harada [12] and depend on soil type. The analytic expressions of the functions 𝐻1and 𝐻2 are given in

[11]. The overall incoherence (incident motion incoherence and soil layer stochasticity) in the spatial variation of the surface

motions is given by:

𝛾𝑐𝑜ℎ(𝜉, 𝜔) = 𝛾𝑏,𝑐𝑜ℎ(𝜉, 𝜔) ∙ 𝛾𝑙,𝑐𝑜ℎ(𝜉, 𝜔) (4)

The 𝛾𝑏,𝑝𝑟𝑜𝑝(𝜉, 𝜔) is a term describing the wave passage effect is given by

𝛾𝑏,𝑝𝑟𝑜𝑝(𝜉, 𝜔) = 𝑒𝑥𝑝{−𝑖𝜔𝜉/𝑐} (5)

To capture the wave passage effect, one needs the information of the apparent wave velocity 𝑣𝑎𝑝𝑝 in the direction of

propagation and the fault-structure geometry. The required parameter 𝑐 is the apparent wave velocity along the structure axis

and can be written as 𝑐 = 𝑣𝑎𝑝𝑝 𝑠𝑖𝑛𝜃⁄ , in which 𝜃 is the angle between the line perpendicular to the structure axis (along a

line connecting all the stations) and the direction of propagation of the incoming waves. The 𝑣𝑎𝑝𝑝 and back-azimuth direction

𝐵𝑎𝑧 (direction of approach of the wavefront) corresponding to the 1980 El-Asnam Earthquake are estimated herein by using

frequency–wave number spectrum analysis. It should be noted that the apparent propagation effects (deterministic phase

variations caused by the wave passage effect) of the surface motions is controlled by that of the incident motion in the

bedrock, since vertical propagation is considered within the layer in the Zerva and Harada model.

3. Abrahamson’s Method The Abrahamson [3] method assumes that the Fourier amplitude of the motion at each station is equal to the Fourier

amplitude of the given reference motion. The difference in the Fourier phase between two stations is due to stochastic

variation in the phase, which can be quantified by lagged coherency function. Derived from the relation between the lagged

coherency and statistical properties of the Fourier phase angles, the coherent ground motion generation is only adjusting

phase angle to match the specified lagged coherency function. In this method the Fourier phase angle of a given reference

motion are used instead of an arbitrary random phase angle. The wave passage effect is included by applying systematic

linear phase shifts in the frequency domain to the stationary time histories generated by this method. More details are given

in [3].

To model the temporal variation of the simulated ground motions, the simulated stationary time histories are multiplied

by the Jennings et al. [15] envelope function, with 𝑡0 and 𝑡𝑛 were used, so that the flat part of the envelop function, or,

equivalently, the strong motion duration of the series, is 𝑡𝑛 − 𝑡0 .

4. Numerical Example A numerical example is presented to simulate non-stationary ground motion time histories using the Abrahamson

algorithm at different locations on the ground surface of a stochastic site with multiple soil layers. For this we consider the

Sogedia site in El-Asnam city. The soil conditions at the site are shown in Figs. 1-(a) and 1-(b), over a length of 1200m. The

material properties are given in the below Fig. 1-(a) and Table 1 in [11], respectively, represent mean values (expected

values). Due to lack of data, it is assumed that stochasticity in soil characteristics is due to the variability in the depth of the

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layers. A Gaussian distribution is assumed for layer thickness with a 20% coefficient of variation. The bedrock is assumed

to be rigid. The ground is uniformly divided into sixty 20m sections.

In the present example of the Sogedia soil profil, we generate transverse component of accelerations on ground surface

at five (n = 5) locations (stations), namely S(0), S(1) , S(2), S(3) and S(4). Station S

(0), which corresponds to the projection

on the surface of the station S′(0) at the rock, is considered as the reference station and located at X=0 in the Fig. 1-(a); the

other stations are separated from it by 40 m, 100 m, 200 m, and 500 m. Epicentral distance of the surface reference station

is 5 km. Simulation of ground motion and estimation of all lagged coherencies were performed by computer codes developed

by the author.

(a)

(b) Fig. 1: (a) Site profile at Sogedia Factory in northwest Algeria with mean layer thicknesses, and (b) a realization of the soil profile

(bedrock not shown) with stochastic layer thickness.

Using the estimated parameters values given in [11] of 𝛼 = 5.87 10−5, 𝜆 = 1.52 (of 𝛾𝑏,𝑐𝑜ℎ(𝜉, 𝜔) in Eq. 2) and 𝜔0 =9.6𝜋 𝑟𝑎𝑑/𝑠 (𝑓0 = 4.8 𝐻𝑧), 𝜎𝜔𝜔 = 0.06, 𝑅𝜔𝜔(𝜉), 𝜁0 = 5% and 𝛽 = 4 𝜋⁄ (of 𝛾𝑙,𝑐𝑜ℎ(𝜉, 𝜔) in Eq. 3), the overall lagged

coherency 𝛾𝑐𝑜ℎ(𝜉, 𝜔) (Eq. 4) at the surface was estimated, and is presented in Fig. 2. The lagged coherency functions exhibit

a sharp decrease near the fundamental frequency of the site 𝜔0 that is caused by the random variability around its mean value

for the different “soil columns” at the site.

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Fig. 2: Lagged coherency function at the surface; obtained by Eq. (4) from bedrock coherency function

(Hindy and Novak model, Eq. 2) and the contribution of stochastic site (Eq. 3).

In this study, a bedrock multiple-station 2-D generic array centred 5 km N265°W of the epicentre of the October 10

1980 𝑀𝑠 = 7.2 Earthquake occurred in El-Asnam region (North-West Algeria) (see Fig. 2 in [9]), is considered to simulate

at 37 stations the NS component time-series using the finite-source Empirical Green’s Function (EGF) method of Irikura et

al. [10]. This simulation is carried out by the widely used finite-source

Fig. 3: Stacked slowness spectrum for N-S component. 𝑆𝑥 and 𝑆𝑦 indicate slowness in the E-W and N-S directions respectively. The F-

K spectral values computed in the frequency range of 0–20 Hz are normalized by the largest value obtained for each frequency. The

estimated direction to the source (back-azimuth direction) is indicated by the black arrow. The corresponding apparent propagation

velocity and the back-azimuth are printed on the title of the plot.

ground motion simulation approach, the so-called Empirical Green’s Function (EGF) method. Based on a frequency–wave

number spectrum (F-K) and the stacked slowness spectra (SS) [16] analysis applied to the N-S component, apparent velocity

𝑣𝑎𝑝𝑝 and back-azimuth direction (clockwise from north) 𝐵𝑎𝑧 of the single plane wave dominating the motions are identified

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and estimated to 𝑣𝑎𝑝𝑝 = 3.26 𝑘𝑚/𝑠 and 𝐵𝑎𝑧 = 710. The stacked slowness spectrum for the NS component is shown in

Fig. 3. The site’s X-axis, containing the stations in Fig. 1-(b), is oriented with an angle of 2650 (clockwise from north) and

using the two estimated values of 𝑣𝑎𝑝𝑝 and 𝐵𝑎𝑧, the required apparent velocity 𝑐 (𝑐 = 𝑣𝑎𝑝𝑝 𝑠𝑖𝑛𝜃⁄ ) along the x-axis to

introduce the wave passage effect (Eq. 5) in the simulation is then equal to 3.36 𝑘𝑚/𝑠.

In addition of the overall lagged coherency 𝛾𝑐𝑜ℎ(𝜉, 𝜔) and wave passage 𝛾𝑏,𝑝𝑟𝑜𝑝(𝜉, 𝜔) functions, the reference surface

motion at the station S(0) needed in the above Abrahamson method of simulation is obtained by convolution at the top of the

site of the generated motion at the rock station S′(0) by [9] using the equivalent linear transfer function of the site estimated

in [11].The parameters of the Jennings envelope function are with 𝑡0 = 2𝑠 and 𝑡𝑛 = 15𝑠 given in [9]. The generated

transverse surface motions at the five stations using the Abrahamson method are shown in Fig. 4. The corresponding 5%

damped acceleration response spectra and arias intensity [17] are shown in Fig. 5. Figure 6 is the comparison represented in

the hyperbolic arctangent (tanh−1) space of the lagged coherencies computed by means of signal processing (as described

in Sect. 2 in [9]) between the generated ground motion time histories (Fig. 4) and the prescribed model (Eq. 4, Fig. 2).

Fig. 4: Transverse component of ground acceleration simulated at the five stations. Time window between the red lines are used to

compute lagged coherencies shown in Fig. 6.

As we can see in the above figures, the simulated spatially variable surface ground-motion time histories at different

locations conserve the same energy and their response spectra (i.e. power spectral densities) are close to that of the reference

motion, and therefore the assumption of the considered homogeneous soil site is preserved by the used Abrahamson method.

A good match can also be observed between the coherency loss functions computed from the simulated motions (Fig. 4) and

the overall incoherence model 𝛾𝑐𝑜ℎ(𝜉, 𝜔) (Eq. 4). It is then proven that the simulated spatial ground motion time histories

are compatible with the target model of coherency loss function. In the absence of data record, the generated time histories

can be used as inputs to multiple supports of long span structures crossing random site.

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Fig. 5: Response spectra (right) and normalized arias intensity (left) of generated motions

at different stations shown in Fig. 4,

Fig. 6: Comparison of coherency loss computed between the generated surface accelerations

with model coherency loss function (Eq. 4, Fig. 2).

5. Conclusions This paper presents a simulation of spatially varying earthquake ground motions on surface of a typical random site in

the epicentral area of the 1980 El-Asnam Earthquake in northwest Algeria. The study is based on lagged coherency of

bedrock ground motion estimated from ground motion field simulated by using the Empirical Green’s Function method, the

contribution of random homogeneous site to the surface lagged coherency and the apparent wave velocity of shear waves

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across the study area during this earthquake. The common technique of Abrahamson for generating spatially variable ground

motions is used considering incoherence, wave passage and site effects. The generated time histories conserve the same

energy at various points on ground surface of a random site and therefore are compatible with the homogeneity assumption

of the site. The simulated spatial ground motions are also compatible with the prescribed model of coherency loss function.

In the absence of data record, the generated time histories can be used as inputs to multiple supports of long span structures

crossing random site.

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